Properties

Label 2-300-1.1-c1-0-1
Degree $2$
Conductor $300$
Sign $1$
Analytic cond. $2.39551$
Root an. cond. $1.54774$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

Related objects

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  + 3-s − 7-s + 9-s + 6·11-s + 5·13-s − 6·17-s + 5·19-s − 21-s − 6·23-s + 27-s − 6·29-s − 31-s + 6·33-s + 2·37-s + 5·39-s − 43-s + 6·47-s − 6·49-s − 6·51-s − 12·53-s + 5·57-s − 6·59-s − 13·61-s − 63-s + 11·67-s − 6·69-s + 2·73-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.377·7-s + 1/3·9-s + 1.80·11-s + 1.38·13-s − 1.45·17-s + 1.14·19-s − 0.218·21-s − 1.25·23-s + 0.192·27-s − 1.11·29-s − 0.179·31-s + 1.04·33-s + 0.328·37-s + 0.800·39-s − 0.152·43-s + 0.875·47-s − 6/7·49-s − 0.840·51-s − 1.64·53-s + 0.662·57-s − 0.781·59-s − 1.66·61-s − 0.125·63-s + 1.34·67-s − 0.722·69-s + 0.234·73-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(300\)    =    \(2^{2} \cdot 3 \cdot 5^{2}\)
Sign: $1$
Analytic conductor: \(2.39551\)
Root analytic conductor: \(1.54774\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{300} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 300,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.627916149\)
\(L(\frac12)\) \(\approx\) \(1.627916149\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 - T \)
5 \( 1 \)
good7 \( 1 + T + p T^{2} \)
11 \( 1 - 6 T + p T^{2} \)
13 \( 1 - 5 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 - 5 T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 + T + p T^{2} \)
47 \( 1 - 6 T + p T^{2} \)
53 \( 1 + 12 T + p T^{2} \)
59 \( 1 + 6 T + p T^{2} \)
61 \( 1 + 13 T + p T^{2} \)
67 \( 1 - 11 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 + 6 T + p T^{2} \)
89 \( 1 + p T^{2} \)
97 \( 1 + 7 T + p T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.66009069683484336461269580959, −10.93906516387019902338031029709, −9.496634055474639300529104381989, −9.108883986383451182036456380552, −8.026266654629362330508166642807, −6.77234120616327244201676192765, −6.03058890823729073510214550931, −4.24382547629296535183762439094, −3.43647697037916365579697730592, −1.63891885984744417728938078317, 1.63891885984744417728938078317, 3.43647697037916365579697730592, 4.24382547629296535183762439094, 6.03058890823729073510214550931, 6.77234120616327244201676192765, 8.026266654629362330508166642807, 9.108883986383451182036456380552, 9.496634055474639300529104381989, 10.93906516387019902338031029709, 11.66009069683484336461269580959

Graph of the $Z$-function along the critical line